A={x||2x-1|>1}B={y|y=|logaX|,X∈[m,n]},B是A的补集(R是全集),n-m的最小值是1/2,求a的值

Question

A={x||2x-1|>1} B={y|y=|logaX|,X∈[m,n]},B是A的补集(R是全集),n-m的最小值是1/2,求a的值

Answer 1

A={x||2x-1|>1} , X>1, 或 X<0
A的补集为 B={y| 0<=y<=1}
B={y|y=|logaX|, X∈[m,n]} = {y|,0<=y<=1},
0<= |logaX| <= 1
0<= logaX <= 1 或 -1 <= logaX <= 0
当 0 当 a>1时, m=1<=X<=a=n 或 m=1/a<=X<=1=n
n-m = 1-a =1/2, a=1/2, 满足0 n-m = 1/a-1=1/2, a=2/3, 满足0 n-m = a-1=1/2, a=3/2, 满足a>1
n-m = 1-1/a=1/2, a=2,满足 a>1